document.write( "Question 753021: x^4+(sqrt 76)x^2-76=0 \n" ); document.write( "
Algebra.Com's Answer #458191 by DrBeeee(684)\"\" \"About 
You can put this solution on YOUR website!
Giving you some tricky questions are they? I think we can handle it. It just a little complicated on-line.
\n" ); document.write( "First, to reduce the order of the polynomial, let
\n" ); document.write( "(1) y = x^2, then your equation reduces to
\n" ); document.write( "(2) y^2 + sqrt(76)*y -76 = 0
\n" ); document.write( "Now use the quadratic equation to find the roots of (2).
\n" ); document.write( "(3) y = \"-sqrt%2876%29%2F2+%2B-+%281%2F2%29%2Asqrt%2876+-4%2A1%2A%28-76%29%29\" or
\n" ); document.write( "(4) y = \"-sqrt%2876%29%2F2+%2B-+%281%2F2%29%2Asqrt%2876+%2B+4%2A76%29\" or
\n" ); document.write( "(5) y = \"-sqrt%2876%29%2F2+%2B-+%281%2F2%29%2Asqrt%2876%29%2A%28sqrt%285%29%29\" or
\n" ); document.write( "(6) y = \"%28sqrt%2876%29%2F2%29%2A%28-1+%2B-+sqrt%285%29%29\"
\n" ); document.write( "Since y = x^2 we get
\n" ); document.write( "(7) x = +/-sqrt(y) and since y has two roots we will get four roots for x.
\n" ); document.write( "Taking the square root of (6) gives us two roots of x
\n" ); document.write( "(8) \"x1+=+%2Bsqrt%28%28sqrt%2876%29%2F2%29%2A%28-1%2Bsqrt%285%29%29%29\" and
\n" ); document.write( "(9) \"x2+=+-sqrt%28%28sqrt%2876%29%2F2%29%2A%28-1%2Bsqrt%285%29%29%29\"
\n" ); document.write( "Likewise we can write the other two roots of x as
\n" ); document.write( "(10) \"x3+=+%2Bsqrt%28%28sqrt%2876%29%2F2%29%2A%28-1-sqrt%285%29%29%29\" and
\n" ); document.write( "(11) \"x4+=+-sqrt%28%28sqrt%2876%29%2F2%29%2A%28-1-sqrt%285%29%29%29\"
\n" ); document.write( "We acn check these roots by substitution into your original equation in x. I'll use x of (8) to show that
\n" ); document.write( "(12) \"x%5E4+%2B+sqrt%2876%29%2Ax%5E2+-76+=+0\"
\n" ); document.write( "We have
\n" ); document.write( "(13) \"x%5E2+=+sqrt%2876%29%2F2%2A%28-1%2Bsqrt%285%29%29\" and
\n" ); document.write( "(14) \"x%5E4+=+76%2F4%2A%28-1%2Bsqrt%285%29%29%5E2\"
\n" ); document.write( "Now put (13) and (14) into (12) and get
\n" ); document.write( "(15) or
\n" ); document.write( "(16) \"76%2F4%2A%28-1%2Bsqrt%285%29%29%5E2+%2B+76%2F2%2A%28-1%2Bsqrt%285%29%29+-+76+=+0\"
\n" ); document.write( "Multiply (16) by 4/76 and get
\n" ); document.write( "(17) \"%28-1%2Bsqrt%285%29%29%5E2+%2B+2%2A%28-1%2Bsqrt%285%29%29+-+4+=+0\" or
\n" ); document.write( "(18) \"%28-1%2Bsqrt%285%29%29%2A%28-1%2Bsqrt%285%29%2B2%29+-+4+=+0\" or
\n" ); document.write( "(19) \"%28-1%2Bsqrt%285%29%29%2A%28%2B1%2Bsqrt%285%29%29+-+4+=+0\" or
\n" ); document.write( "(20) \"-1%2Bsqrt%285%29%5E2+-+4+=+0\" or
\n" ); document.write( "(21) \"-1+%2B+5+-+4+=+0\" or
\n" ); document.write( "(22) 0 = 0 check
\n" ); document.write( "Answers: the roots of your quardic equation are given by (8)-(11)\r
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